A hereditarily indecomposable $ {\mathcal{L}_{\infty}} $-space that solves the scalar-plus-compact problem

Abstract

We construct a hereditarily indecomposable Banach space with dual space isomorphic to ℓ1. Every bounded linear operator on this space is expressible as λI + K with λ a scalar and K compact.

Rota's universal operators and invariant subspaces in Hilbert spaces

“…Moreover, σ(0) is not on the curve σ(Γ). Proof of [1]: The map z → 1/(z + 1) takes the unit disk onto the half plane {w : Re (w) > 1/2} ⊂ {w : −π/2 < arg(w) < π/2} with the curve Γ going to the line Re (w) = 1/2, where {e iθ : −π < θ < 0} is mapped to {1/2 + iy : y > 0} and the other half of the circle to the other half of the line. This means z → 1/ √ z + 1 maps the disk into a subset of {w : −π/4 < arg(w) < π/4} and the point x + iy is in the image of the disk under this map if and only if (x + iy) 2 satisfies Re (x + iy) 2 > 1 2 .…”

“…More recently, a rather striking affirmative result in the Banach space context has been provided by Argyros and Hydon [1], who constructed an infinite dimensional Banach space such that every linear bounded operator is a compact perturbation of a scalar operator. Therefore, by Lomonosov's Theorem, every linear bounded operator in this Banach space has a nontrivial closed hyperinvariant subspace.…”

“…In 1985, Read [26] constructed a bounded linear operator without nontrivial closed invariant subspaces in the well-known sequence space 1 . His construction, which appears to be the first example of such an operator on any of the classical Banach spaces, was simpler than Enflo's in some sense.…”

Shift invariant preduals of ℓ 1(ℤ)

“…A similar argument holds if x ∈ σ * (F ) \ F . Thus (1) implies (2). Conversely, when (2) holds, let (a α ) be a weak * -null net in 1 (Z) and let x ∈ F .…”

“…In [8], a predual Y of 1 (Z) was constructed such that Y has the Radon-Nikodym property and each infinite-dimensional subspace of Y contains a further infinite-dimensional subspace which is reflexive. This construction was an inspiration for the recent solution to the scalar-compact problem [2]: this exotic Banach space is also an 1 predual. Indeed, in [15], it is shown that if X is any Banach space with separable dual, then there is an 1 predual E which contains an isomorphic copy of X.…”